Area integral estimates for caloric functions

Brown, Russell M.

doi:10.1090/S0002-9947-1989-0994163-7

Area integral estimates for caloric functions
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by Russell M. Brown

Trans. Amer. Math. Soc. 315 (1989), 565-589

DOI: https://doi.org/10.1090/S0002-9947-1989-0994163-7

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Abstract:

We study the relationship between the area integral and the parabolic maximal function of solutions to the heat equation in domains whose boundary satisfies a $\left ({\frac {1}{2},1}\right )$ mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in ${L^p}(\mu )$, $0 < p < \infty$. The measure $\mu$ must satisfy Muckenhoupt’s ${A_\infty }$-condition with respect to caloric measure. We also give a Fatou theorem which shows that the existence of parabolic limits is a.e. (with respect to caloric measure) equivalent to the finiteness of the area integral.

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